| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Increasing/decreasing intervals |
| Difficulty | Moderate -0.8 This is a straightforward C2 differentiation question requiring basic power rule application (converting √x to x^(1/2)) and interpreting dy/dx < 0. Part (a) is pure recall, part (b) requires substitution and solving a simple inequality—easier than average A-level questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 2x - \frac{1}{2}kx^{-\frac{1}{2}}\) | M1 A1 | Having an extra term e.g. \(+C\) is A0. M: \(x^2 \to cx\) or \(k\sqrt{x} \to cx^{-\frac{1}{2}}\) (\(c\) constant, \(c \neq 0\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substituting \(x=4\) into their \(\frac{dy}{dx}\) and comparing with zero | M1 | Allowed for \(<, >, =, \leq, \geq\). Substitution of \(x=4\) into \(y\) scores M0. \(\frac{dy}{dx}=0\) may be implied for M1 when a value of \(k\) or inequality is found |
| \(8 - \frac{k}{4} < 0 \Rightarrow k > 32\) | A1 | Correct inequality needed. Working must be seen; \(k>32\) alone is M0 A0 |
## Question 3:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 2x - \frac{1}{2}kx^{-\frac{1}{2}}$ | M1 A1 | Having an extra term e.g. $+C$ is A0. M: $x^2 \to cx$ or $k\sqrt{x} \to cx^{-\frac{1}{2}}$ ($c$ constant, $c \neq 0$) |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substituting $x=4$ into their $\frac{dy}{dx}$ and comparing with zero | M1 | Allowed for $<, >, =, \leq, \geq$. Substitution of $x=4$ into $y$ scores M0. $\frac{dy}{dx}=0$ may be implied for M1 when a value of $k$ or inequality is found |
| $8 - \frac{k}{4} < 0 \Rightarrow k > 32$ | A1 | Correct inequality needed. Working must be seen; $k>32$ alone is M0 A0 |
---3.
$$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Given that $y$ is decreasing at $x = 4$, find the set of possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2010 Q3 [4]}}